I have 15 GPU WUs in inventory, each of which will take about 00:05:33 to complete. Yet I constantly get "No work sent" response from a server that has 92,000 WUs available. No explanation is forthcoming about why no work sent. Computer ID: 6120747. Why is that?

I'm not sure where you're seeing 92,000 workunits available, but when I check I see 3 workunits available. Since the server status page isn't updated in real time, those three workunits may not actually be there by the next BOINC request.